Modular representations and the homotopy of low rank p-local CW-complexes

Fix an odd prime p and let X be the p-localization of a finite suspended CW-complex. Given certain conditions on the reduced mod-p homology H˜?(X;Zp) of X, we use a decomposition of ??X due to the second author and computations in modular representation theory to show there are arbitrarily large integers i such that ?? i X is a homotopy retract of ??X. This implies the stable homotopy groups of ?X are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for ?X. Under additional assumptions on H˜?(X;Zp), we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of ??X that has infinitely many finite H-spaces as factors